Contemporary Mathematics

Volume 178, 1994

On h-Vectors and Symmetry

Ron M. Adin

Abstract

Tight lower bounds are obtained for all the face-numbers of a rational simpli-

cial polytope that admits a fixed-point-free linear symmetry of prime-power

order. These bounds are best possible for all compatible values of the poly-

tope's dimension, its number of vertices, and the order of its given symmetry.

They are consequences of corresponding bounds on the h-vector of the poly-

tope, and extend previous results of Stanley and of the author. Also included

is a short survey of some concepts and results in the combinatorial theory of

group actions on polytopes.

1 Statement of Results

The purpose of this paper is, primarily, to give the details of a certain result about

polytopes with a given symmetry. We also take this opportunity to include a brief

survey of terminology and results concerning group actions on polytopes. Therefore,

this opening section contains complete statements of the main results, whereas all

the background material, including definitions of some terms used in the statement

of the results, is deferred to Section 2. Details about the organization of the rest of

the paper may be found at the end of the current section.

Theorem 1

Let P be

a

simplicial

convex

d-polytope with rational vertices, admitting

a

fixed-

point-free linear action of

a

cyclic group G of order n, where n

=

p" is

a

prime

power. The dimension d is necessarily divisible by ¢(

n) =

p"-

1

(p - 1), where ¢

is Euler's totient function. Let

Pmin(G,

d) be the free sum of dj(p-

1)

copies of

the (p- 1)-simplex, and let the difference between the h-polynomials of P and of

Pmin(G,

d) be

d

hp(q)- hpmin(G,d)(q)

= L:iqi.

(1)

i=O

1991 Mathematics Subject Classification. Primary 52B05; Secondary 05E25, 14M25, 52B15,

52B20.

Research supported in part by the Israel Science Foundation, administered by the Israel

Academy of Sciences and Humanities.

1

©

1994 American Mathematical Society

0271-4132/94 $1.00

+

$.25 per page

http://dx.doi.org/10.1090/conm/178/01889